MinT - Best Known (116, 116+39, s)-Nets in Base 4

Best Known (116, 116+39, s)-Nets in Base 4

(116, 116+39, 531)-Net over F₄ — Constructive and digital

Digital (116, 155, 531)-net over F₄, using

t-expansion [i] based on digital (115, 155, 531)-net over F₄, using
- 7 times m-reduction [i] based on digital (115, 162, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 54, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(116, 116+39, 576)-Net in Base 4 — Constructive

(116, 155, 576)-net in base 4, using

t-expansion [i] based on (115, 155, 576)-net in base 4, using
- 1 times m-reduction [i] based on (115, 156, 576)-net in base 4, using
  - trace code for nets [i] based on (11, 52, 192)-net in base 64, using
    - 4 times m-reduction [i] based on (11, 56, 192)-net in base 64, using
      - base change [i] based on digital (3, 48, 192)-net over F₁₂₈, using
        net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(116, 116+39, 1449)-Net over F₄ — Digital

Digital (116, 155, 1449)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(116, 116+39, 200408)-Net in Base 4 — Upper bound on s

There is no (116, 155, 200409)-net in base 4, because

1 times m-reduction [i] would yield (116, 154, 200409)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 521 516082 640767 504931 366389 293718 643542 636869 052888 737622 685069 586982 502869 687308 855913 848872 > 4¹⁵⁴ [i]